| Uniqueness of large invariant measures for Z k actions with Cartan homotopy data (2009) | |||||||||||||||
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| ABSTRACT. Every C 2 action α of Z k, k ≥ 2, on the (k+ 1)-dimensional torus whose elements are homotopic to the corresponding elements of an action α0 by hyperbolic linear maps has exactly one invariant measure that projects to Lebesgue measure under the semiconjugacy between α and α0. This measure is absolutely continuous and the semiconjugacy provides a measure-theoretic isomorphism. The semiconjugacy has certain monotonicity properties and preimages of all points are connected. There are many periodic points for α for which the eigenvalues for α and α0 coincide. We describe some nontrivial examples of actions of this kind. 1. PRELIMINARIES 1.1. Introduction. This paper constitutes a direct continuation of [3]. We shall use the terminology and results from [3] with only occasional references. Let α0 be a Z k Cartan action on T k+1 and let α be a smooth Z k action whose elements are homotopic to the corresponding elements of α0. 1 “Smooth ” in our context means C 2 although most of the arguments hold for C 1+ε actions for any | |||||||||||||||
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